Preparation of the martini generally consists of a small amount of vermouth poured over ice to slightly melt and flavor the ice. The desired amount of gin or vodka is added and the mixture is shaken or stirred according to taste and then served without the ice.
Phillip Boyce, after being summoned to Captain Christopher Pike's quarters, noted that no one wants a warm martini. When asked why he thought Pike needed one, Boyce commented, "Sometimes a man will tell his bartender things he'll never tell his doctor." (TOS: "The Cage", "The Menagerie, Part I")
Vic's Las Vegas Lounge serves martinis. When Jake Sisko and Kesha joined Nog at Vic's, she ordered a martini. Julian Bashir ordered a vodka martini, stirred not shaken. (DS9: "It's Only a Paper Moon", "Badda-Bing, Badda-Bang")
Also see: Klingon martini